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     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
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     "source": [
      "# MTH 9879 Market Microstructure Models,    Spring 2015 \n",
      "\n",
      "## Lecture 7: Long memory of order flow and market impact\n",
      "\n",
      "Tai-Ho Wang    \n",
      "Department of Mathematics   \n",
      "  \n",
      "  \n",
      "<h2><img src=\"http://mfe.baruch.cuny.edu/wp-content/uploads/2014/09/BCCUNYstacked_BLK.jpg\" align = \"center\" width=160></h2>  \n",
      "\n",
      "$$\n",
      "\\newcommand{\\bea}{\\begin{eqnarray}}\n",
      "\\newcommand{\\eea}{\\end{eqnarray}}\n",
      "\\newcommand{\\supp}{\\mathrm{supp}}\n",
      "\\newcommand{\\F}{\\mathcal{F} }\n",
      "\\newcommand{\\cF}{\\mathcal{F} }\n",
      "\\newcommand{\\E}{\\mathbb{E} }\n",
      "\\newcommand{\\Eof}[1]{\\mathbb{E}\\left[ #1 \\right]}\n",
      "\\def\\Cov{{ \\mbox{Cov} }}\n",
      "\\def\\Var{{ \\mbox{Var} }}\n",
      "\\newcommand{\\1}{\\mathbf{1} }\n",
      "\\newcommand{\\p}{\\partial}\n",
      "\\newcommand{\\PP}{\\mathbb{P} }\n",
      "\\newcommand{\\Pof}[1]{\\mathbb{P}\\left[ #1 \\right]}\n",
      "\\newcommand{\\QQ}{\\mathbb{Q} }\n",
      "\\newcommand{\\R}{\\mathbb{R} }\n",
      "\\newcommand{\\DD}{\\mathbb{D} }\n",
      "\\newcommand{\\HH}{\\mathbb{H} }\n",
      "\\newcommand{\\spn}{\\mathrm{span} }\n",
      "\\newcommand{\\cov}{\\mathrm{cov} }\n",
      "\\newcommand{\\HS}{\\mathcal{L}_{\\mathrm{HS}} }\n",
      "\\newcommand{\\Hess}{\\mathrm{Hess} }\n",
      "\\newcommand{\\trace}{\\mathrm{trace} }\n",
      "\\newcommand{\\LL}{\\mathcal{L} }\n",
      "\\newcommand{\\s}{\\mathcal{S} }\n",
      "\\newcommand{\\ee}{\\mathcal{E} }\n",
      "\\newcommand{\\ff}{\\mathcal{F} }\n",
      "\\newcommand{\\hh}{\\mathcal{H} }\n",
      "\\newcommand{\\bb}{\\mathcal{B} }\n",
      "\\newcommand{\\dd}{\\mathcal{D} }\n",
      "\\newcommand{\\g}{\\mathcal{G} }\n",
      "\\newcommand{\\half}{\\frac{1}{2} }\n",
      "\\newcommand{\\T}{\\mathcal{T} }\n",
      "\\newcommand{\\bit}{\\begin{itemize}}\n",
      "\\newcommand{\\eit}{\\end{itemize}}\n",
      "\\newcommand{\\beq}{\\begin{equation}}\n",
      "\\newcommand{\\eeq}{\\end{equation}}\n",
      "\\newcommand{\\tr}{\\mbox{tr}}\n",
      "\\newcommand{\\underbar}{\\underline}\n",
      "$$\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "## Outline of Lecture 7\n",
      "\n",
      "-   Long memory of order flow\n",
      "\n",
      "-   Definitions of market impact\n",
      "\n",
      "-   Why market impact is a concave function of volume.\n",
      "\n",
      "-   Two models of market impact:\n",
      "\n",
      "    -   Permanent, state-dependent impact\n",
      "\n",
      "    -   Transient impact\n",
      "\n",
      "-   Equivalence of these two models\n",
      "\n",
      "-   Including limit orders and cancelations\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Long memory of order flow\n",
      "\n",
      "-   Prices evolve as a function of order flow and the arrival of new\n",
      "    orders in response to that order flow.\n",
      "\n",
      "-   Price dynamics (especially dynamics of the mid-quote) are\n",
      "    well-described by Brownian motion.\n",
      "\n",
      "-   As we have seen already, order flow is a highly autocorrelated\n",
      "    long-memory process.\n",
      "\n",
      "-   We know that the price process is reasonably efficient\n",
      "\n",
      "    -   That is price changes are almost uncorrelated which implies that\n",
      "        variance grows approximately linearly in trading time.\n",
      "\n",
      "-   It follows that the market response to order flow must strongly\n",
      "    depend on the past history of order flow.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Long memory of order flow\n",
      "\n",
      "<h2><img src=\"http://mfe.baruch.cuny.edu/wp-content/uploads/2015/03/Figure1.png\" align = \"center\" width=600></h2>\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Autocorrelation of MSFT trade signs"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%load_ext rmagic"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%R\n",
      "download.file(url=\"http://mfe.baruch.cuny.edu/wp-content/uploads/2015/03/MSFT130311.rData_.zip\", destfile=\"MSFT130311.zip\")\n",
      "unzip(zipfile=\"MSFT130311.zip\")\n",
      "load(\"MSFT130311.rData\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "trying URL 'http://mfe.baruch.cuny.edu/wp-content/uploads/2015/03/MSFT130311.rData_.zip'\n",
        "Content type 'application/zip' length 71919 bytes (70 Kb)\n",
        "opened URL\n",
        "==================================================\n",
        "downloaded 70 Kb\n",
        "\n"
       ]
      }
     ],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%R\n",
      "\n",
      "# Plot tradesigns\n",
      "ts.both <- sign(msft130311.both$signed.shares)\n",
      "ts.bats <- sign(msft130311.bats$signed.shares)\n",
      "ts.inet <- sign(msft130311.inet$signed.shares)\n",
      "\n",
      "# Plot autocorrelation function of trade signs\n",
      "acts.bats <- acf(ts.bats,plot=F)\n",
      "acts.inet <- acf(ts.inet,plot=F)\n",
      "acts.both <- acf(ts.both,plot=F)\n",
      "\n",
      "# AC computations\n",
      "lag.bats <- acts.bats$lag[-1]\n",
      "acf.bats <- acts.bats$acf[-1]\n",
      "\n",
      "lag.inet <- acts.inet$lag[-1]\n",
      "acf.inet <- acts.inet$acf[-1]\n",
      "\n",
      "lag.both <- acts.both$lag[-1]\n",
      "acf.both <- acts.both$acf[-1]\n",
      "\n",
      "# ACF plot\n",
      "plot(lag.both,acf.both,type=\"l\",xlab=expression(paste(\"Lag \",tau)),ylab=expression(paste(\"ACF \", rho(tau))), col=\"dark green\",ylim=c(0,1))\n",
      "lines(lag.bats,acf.bats,type=\"l\",xlab=\"log(lag)\",ylab=\"log(ACF)\",col=\"red\")\n",
      "lines(lag.inet,acf.inet,type=\"l\",xlab=\"log(lag)\",ylab=\"log(ACF)\",col=\"blue\")\n",
      "legend(\"topright\", c(\"BATS     \",\"INET\",\"Both\"),col=c(\"red\",\"blue\",\"dark green\"),lty=1, inset=0.02)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
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i8GAMGoRXXik7W686Odk7je8xfnyP8dduXTt49uD+M/v/9/v/DmUfunH7hmc9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\n"
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Data thanks to Alejandro Ca\u00f1ete is from 11-Mar-2013."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### MSFT autocorrelation log-log plot"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%R\n",
      "\n",
      "# Log-log plot\n",
      "plot(log(lag.both),log(acf.both),xlab=\"log(lag)\",ylab=\"log(ACF)\", col=\"dark green\",ylim=c(-2,.5))\n",
      "(fit.both <- lm(log(acf.both)[2:20]~log(lag.both)[2:20]))\n",
      "abline(fit.both,col=\"dark green\");\n",
      "slope.both <- fit.both$coef[2];\n",
      "\n",
      "points(log(lag.bats),log(acf.bats),xlab=\"log(lag)\",ylab=\"log(ACF)\", col=\"red\")\n",
      "(fit.bats <- lm(log(acf.bats)[2:20]~log(lag.bats)[2:20]))\n",
      "abline(fit.bats,col=\"red\");\n",
      "slope.bats <- fit.bats$coef[2];\n",
      "\n",
      "points(log(lag.inet),log(acf.inet),xlab=\"log(lag)\",ylab=\"log(ACF)\", col=\"blue\")\n",
      "(fit.inet <- lm(log(acf.inet)[2:20]~log(lag.inet)[2:20]))\n",
      "abline(fit.inet,col=\"blue\");\n",
      "slope.inet <- fit.inet$coef[2];\n",
      "\n",
      "leg1 <- paste(\"BATS: Slope\",format(slope.bats, digits = 4),\"              \")\n",
      "leg2 <- paste(\"INET: Slope\",format(slope.inet, digits = 4))\n",
      "leg3 <- paste(\"Both: Slope\",format(slope.both, digits = 4))\n",
      "\n",
      "legend(\"bottomleft\", c(leg1,leg2,leg3),col=c(\"red\",\"blue\",\"dark green\"),lty=1, inset=0.02)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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sqKiYmJhYvhS6URZHCKyscOYMdu6EoyM+fxYXB5OJhAT06IFKNXmlBPgEjs46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\nLfjyS1y4AIMBOTmQSrFvn7+b5abyMovj6689/X8UFYXo6OKdotVqJRK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      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Data thanks to Alejandro Ca\u00f1ete is from 11-Mar-2013."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Long memory processes\n",
      "\n",
      "-   Stochastic processes for which the autocorrelation function decays\n",
      "    asymptotically as a power-law with an exponent smaller than one are\n",
      "    called <span>*long-memory*</span> processes.\n",
      "\n",
      "-   For such a process, $$\\rho(\\tau) \\sim \\frac{L(\\tau)}{\\tau^\\alpha}$$\n",
      "    where $L(\\cdot)$ is a slowly-varying function and $\\alpha <1$.\n",
      "\n",
      "    -   $L(x)$ is a slowly varying function if\n",
      "        $\\lim_{x \\to \\infty} L(t\\,x)/L(x) = 1, \\,\\forall t$.\n",
      "\n",
      "-   Models of long-memory processes include Fractional Brownian Motion\n",
      "    and ARFIMA (sometimes called FARIMA).\n",
      "\n",
      "    -   Autoregressive Fractionally Integrated Moving Average\n",
      "\n",
      "-   Volume and volatility are also both widely believed to be\n",
      "    long-memory processes.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### ARFIMA\n",
      "\n",
      "-   A series $\\{X_t\\}$ is $ARFIMA(p,d,q)$ if the series $(1-B)^d\\,X_t$\n",
      "    is a stationary $ARMA(p,q)$ time series where $B$ is the backward\n",
      "    shift operator.\n",
      "\n",
      "-   The special case $d=0$ gives $ARMA(p,q)$ and the case $d=1$ gives\n",
      "    $ARIMA(p,q)$.\n",
      "\n",
      "\n",
      "-   $$(1-B)^d=1+\\sum_{i=1}^\\infty\\,\\frac{d\\,(d-1)...\\,(d-j+1)}{j!}\\,(-1)^j\\,B^j$$\n",
      "\n",
      "\n",
      "-   When $d< 1/2$, the autocorrelation function\n",
      "    $$\\rho(k)\\sim{k^{2\\,d-1}}\\quad \\text{ as } k \\to \\infty$$\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Unwrapping the definition of ARFIMA\n",
      "\n",
      "-   An $ARMA(p,q)$ model has the form\n",
      "    $$X_t=\\phi_0+\\sum_{i=1}^p\\,\\phi_{i}\\,X_{t-i}+\\eta_t-\\sum_{j=1}^q\\,\\theta_j\\,\\eta_{t-j}$$\n",
      "\n",
      "-   This can be rewritten using the backshift operator $B$ as\n",
      "    $$\\left(1-\\sum_{i=1}^p\\,\\phi_i\\,B^i\\right) X_t=\\phi_0+\\left(1-\\sum_{j=1}^q\\,\\theta_j\\,B^j\\right) \\eta_t$$\n",
      "\n",
      "\n",
      "-   If $X_t$ is $ARIMA(p,q)$ then $Y_t=(1-B) X_t$ is $ARMA(p,q)$. $X_t$\n",
      "    is then \u201cintegrated\u201d $ARMA(p,q)$.\n",
      "\n",
      "-   If $(1-B)^d X_t$ is $ARMA(p,q)$ with $d \\in (-1/2,1/2)$, $X_t$ is\n",
      "    then \u201cfractionally integrated\u201d $ARMA(p,q)$.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### The simplest case: $ARFIMA(0,d,0)$\n",
      "\n",
      "-   In this case, $(1-B)^d\\,X_t=\\eta_t$.\n",
      "\n",
      "    -   $d=0$ gives white noise; $d=1$ gives a random walk.\n",
      "\n",
      "\n",
      "-   Then $X_t$ has the MA representation\n",
      "\n",
      "    <a name=\"eq:arfimaMA\"></a>(1)\n",
      "    $$x_t=\\eta_t+\\sum_{j=1}^\\infty\\,\\psi_j\\,\\eta_{t-j}$$\n",
      "\n",
      "    $$\\text{with }\\psi_j=\\left(\n",
      "    \\begin{array}{c}j+d-1\\\\j\\end{array}\n",
      "    \\right)=\\frac{d\\,(d-1)...\\,(d-j+1)}{j!}$$\n",
      "\n",
      "\n",
      "-   The ACF of $X_t$ is\n",
      "\n",
      "    <a name=\"eq:arfimaACF\"></a>(2)\n",
      "    \\begin{eqnarray*}\n",
      "    \\rho(k)&=&\\frac{\\Gamma(1-2\\,d)}{\\{\\Gamma(1-d)\\}^2}\\,\\frac{d\\,(1+d)...\\,(k-1+d)}{(1-d)\\,(2-d)\\,...\\,(k-d)}\\\\\n",
      "    &\\sim&\\frac{\\Gamma(1-2\\,d)}{\\Gamma(d)\\,\\Gamma(1-d)}\\,k^{2\\,d-1} ~ \\text{ as } k \\to \\infty \\nonumber\n",
      "    \\end{eqnarray*}\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Empirical results\n",
      "\n",
      "-   In an earlier slide, we found $\\rho(\\tau) \\sim \\tau^{-\\alpha}$ with\n",
      "    $\\alpha \\approx 0.6$ for MSFT.\n",
      "\n",
      "-   Lillo and Farmer found $\\alpha \\approx 0.6$ on the LSE.\n",
      "\n",
      "-   Bouchaud et al. found $\\alpha \\in (0.2,0.7)$ for stocks on the Paris\n",
      "    Stock Exchange.\n",
      "\n",
      "\n",
      "-   Note that estimating $\\alpha$ as we just did by fitting a straight\n",
      "    line in a log-log plot of the autocorrelation function is\n",
      "    notoriously inaccurate. Much more dependable techniques are\n",
      "    available.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Explanations for long-memory\n",
      "\n",
      "There are two main explanations:\n",
      "\n",
      "-   Herding behavior by traders.\n",
      "\n",
      "    -   We saw previously that herding can be optimal strategic\n",
      "        behavior. Buy orders follow buy orders etc.\n",
      "\n",
      "    -   There can be two types of herding:\n",
      "\n",
      "        -   Traders respond in the same way to public information.\n",
      "\n",
      "        -   Some traders copy other traders.\n",
      "\n",
      "-   Splitting of large trades.\n",
      "\n",
      "    -   At any given time, there is insufficient depth in the order book\n",
      "        to accommodate a large trade. Such trades need to be split.\n",
      "\n",
      "    -   This is also consistent with the optimal strategy of the\n",
      "        informed trader in the Kyle model.\n",
      "\n",
      "-   [T\u00f3th, Lillo et al.]<sup id=\"cite_ref-TothLillo\" class=\"reference\"><a href=\"#cite_note-TothLillo\"><span>[</span>8<span>]</span></a></sup>\n",
      " argue convincingly that the dominant effect is\n",
      "    order-splitting.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Order splitting\n",
      "\n",
      "-   We call a large order that is yet to be revealed to the market a\n",
      "    <span>*hidden*</span> order or a <span>*meta order*</span>.\n",
      "\n",
      "    -   The trader knows his intentions but others in the market do not.\n",
      "\n",
      "    -   The objective of the trader is to minimize market impact by\n",
      "        hiding his intentions for as long as possible.\n",
      "\n",
      "\n",
      "-   Large institutional trades such as pension fund manager transitions\n",
      "    often take a matter of weeks to complete.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### A simple model for order splitting\n",
      "\n",
      "In this model from [Lillo, Mike and Farmer]<sup id=\"cite_ref-LilloMikeFarmer\" class=\"reference\"><a href=\"#cite_note-LilloMikeFarmer\"><span>[</span>6<span>]</span></a></sup>,\n",
      "\n",
      "\n",
      "\n",
      "-   There are $N$ meta orders in the market whose sizes $V_i$ are drawn\n",
      "    from the distribution $P(V)$.\n",
      "\n",
      "-   Meta order signs $\\epsilon_i$ are randomly chosen.\n",
      "\n",
      "-   At time $t$, an existing order $i$ is chosen with uniform\n",
      "    probability and one unit is traded so that $$V_i(t+1)=V_i(t)-1.$$\n",
      "\n",
      "    -   This generates a child (revealed) order of size $1$ and sign\n",
      "        $\\epsilon_i$.\n",
      "\n",
      "-   A meta order dies if $V_i(t+1)=0$ in which case a new meta order is\n",
      "    generated.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### The Lillo, Mike, Farmer model of order splitting\n",
      "\n",
      "We paraphrase the argument of [Lillo, Mike and Farmer]<sup id=\"cite_ref-LilloMikeFarmer\" class=\"reference\"><a href=\"#cite_note-LilloMikeFarmer\"><span>[</span>6<span>]</span></a></sup>:\n",
      "\n",
      "-   Let $\\epsilon_t$ be the sign of the child order observed at time\n",
      "    $t$. Consider the autocorrelation function\n",
      "    $\\rho(\\tau)=\\langle \\epsilon_t\\,\\epsilon_{t+\\tau} \\rangle$ of order\n",
      "    signs.\n",
      "\n",
      "-   By assumption, if two child orders come from different metaorders,\n",
      "    their order signs are uncorrelated.\n",
      "\n",
      "-   The probability that a child order drawn at random comes from a\n",
      "    metaorder of length $L$ is proportional to $L\\,p(L)$ where $p(L)$ is\n",
      "    the probability that a metaorder has length $L$.\n",
      "\n",
      "-   Conditioning on metaorder length $L$, we may then write\n",
      "    $$\\rho(\\tau)=\\frac{1}{\\bar L}\\,\\sum_{L=1}^\\infty\\,L\\,p(L)\\,q(\\tau|L)$$\n",
      "    where $q(\\tau|L)$ is the probability that two child orders $\\tau$\n",
      "    apart come from the same metaorder of length $L$.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Heuristic derivation of $q(\\tau|L)$\n",
      "\n",
      "-   We have $$q(\\tau|L) = \\frac 1 N \\,w(L,\\tau)$$ where $w(L,\\tau)$ is\n",
      "    the probability that a given order of length $L$ is still active\n",
      "    after time $\\tau$.\n",
      "\n",
      "-   If child orders are executed in strict rotation, an order of\n",
      "    remaining length $\\ell$ will remain active for roughly $\\ell\\,N$\n",
      "    periods.\n",
      "\n",
      "-   After time $\\tau$, only orders of remaining length\n",
      "    $\\ell \\geq \\tau/N$ will remain active.\n",
      "\n",
      "-   Thus\n",
      "    $$w(L,\\tau) = \\frac 1 L \\, \\sum_{\\ell=\\tau/N}^{L-1} 1 = \\frac 1 L \\, \\left(L-1 - \\frac{\\tau}{N}\\right)^+$$\n",
      "    and so\n",
      "    $$q(\\tau|L)=\\frac 1N\\, w(L,\\tau) \\approx \\frac 1N\\,  \\frac 1 L \\, \\left(L-1 - \\frac{\\tau}{N}\\right)^+.$$\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Computation of $\\rho(\\tau)$\n",
      "\n",
      "Then\n",
      "\n",
      "<a name=\"eq:acf\"></a>(3)\n",
      "\\begin{eqnarray*}\n",
      "\\rho(\\tau) &=& \\frac{1}{\\bar L}\\,\\sum_{L=1}^\\infty\\,L\\,p(L)\\,q(\\tau|L)\\nonumber\\\\\n",
      "&\\approx &\\frac{1}{\\bar L}\\, \\int_0^\\infty\\,p(L)\\,\\frac 1N\\,  \\left(L-1 - \\frac{\\tau}{N}\\right)^+\\,dL\\nonumber\\\\\n",
      "&=&\\frac{1}{\\bar L}\\, \\int_{1+\\tau/N}^\\infty\\,\\frac 1N\\, \\left(L-1 - \\frac{\\tau}{N}\\right) \\,p(L)\\,dL.\\end{eqnarray*}\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### The power-law case\n",
      "\n",
      "In the realistic case where metaorder sizes $L$ are power-law\n",
      "distributed with $$p(L) =\\gamma  L^{-(1+\\gamma)},$$ performing the\n",
      "integration [(3)](#eq:acf) explicitly gives\n",
      "$$\\rho(\\tau) =N^{\\gamma -2} (N+\\tau )^{1-\\gamma } \\sim\\left\\{\\begin{array}{cc}\n",
      "  1/N & \\text{ if } \\tau \\ll N \\\\ \n",
      "N^{\\gamma -2} \\,\\tau^{1-\\gamma }  &  \\text{ if } \\tau \\gg N .\n",
      "\\end{array}\\right.$$ In particular, if $\\gamma=3/2$ as is more or less\n",
      "the case empirically for many stocks, we have\n",
      "$$\\rho(\\tau) \\sim \\frac{1}{\\sqrt{\\tau}} \\text{ for large } \\tau.$$\n",
      "\n",
      "-   The LMF model gives a link between the distribution of order sizes\n",
      "    and the autocorrelation function of order signs.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Empirical confirmation\n",
      "\n",
      "-   [T\u00f3th, Lillo et al.]<sup id=\"cite_ref-TothLillo\" class=\"reference\"><a href=\"#cite_note-TothLillo\"><span>[</span>8<span>]</span></a></sup> perform a careful analysis of order flow data from the\n",
      "    London Stock Exchange containing exchange membership identifiers.\n",
      "\n",
      "-   They conclude that order splitting is indeed the dominant cause of\n",
      "    the long memory of the order sign process.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Evidence based on membership codes\n",
      "\n",
      "<h2><img src=\"http://mfe.baruch.cuny.edu/wp-content/uploads/2015/03/Figure2.png\" align = \"center\" width=600></h2>\n",
      "<a name=\"fig2\"></a>"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Evidence based on membership codes\n",
      "\n",
      "-   Assuming meta orders are executed using only a few brokers, we would\n",
      "    expect to see more autocorrelation for a given broker order flow\n",
      "    that for order flow in aggregate.\n",
      "\n",
      "-   [Figure 2](#fig2) from [Bouchaud, Farmer, Lillo]<sup id=\"cite_ref-BouchaudFarmerLillo\" class=\"reference\"><a href=\"#cite_note-BouchaudFarmerLillo\"><span>[</span>2<span>]</span></a></sup> shows just that and is strong\n",
      "    evidence for order splitting being the dominant cause of long memory\n",
      "    in order flow.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### More evidence for order splitting\n",
      "\n",
      "[T\u00f3th, Lillo et al.]<sup id=\"cite_ref-TothLillo\" class=\"reference\"><a href=\"#cite_note-TothLillo\"><span>[</span>8<span>]</span></a></sup> present even more convincing evidence for order-splitting as\n",
      "opposed to herding as the principal explanation for the long memory of\n",
      "order flow.\n",
      "\n",
      "They decompose the sample autocorrelations as follows:\n",
      "\n",
      "-   $\\epsilon^i_t = 1$ means a buy order was placed by investor $i$\n",
      "\n",
      "-   $\\epsilon^i_t = 0$ means an order placed by another investor\n",
      "\n",
      "-   $\\epsilon^i_t =-1$ means a sell order placed by investor $i$.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Decomposition of autocorrelation function\n",
      "\n",
      "Assuming the long-term average order sign is zero, we have\n",
      "$$C(\\tau)=\\langle\\epsilon_t\\,\\epsilon_{t+\\tau}\\rangle=\\frac{1}{N}\\,\\sum_t\\,\\epsilon_{t}\\,\\epsilon_{t+\\tau}\n",
      "=\\frac{1}{N}\\,\\sum_t\\,\\sum_{i,j}\\epsilon^i_{t}\\,\\epsilon^j_{t+\\tau}$$\n",
      "Define the autocorrelation between $i$ and $j$ orders as\n",
      "$$C^{ij}(\\tau)=\\langle\\epsilon^i_{t}\\,\\epsilon^j_{t+\\tau}\\rangle=\\frac{1}{N^{ij}(\\tau)}\\,\\sum_t\\,\\epsilon^i_{t}\\,\\epsilon^j_{t+\\tau}$$\n",
      "and let $\\theta^{ij}(\\tau) = N^{ij}(\\tau)/N$ be the fraction of times\n",
      "that an order from investor $i$ at time $t$ is followed by an order from\n",
      "investor $j$ at time $t + \\tau$."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "Then\n",
      "\n",
      "\\begin{eqnarray*}\n",
      "C(\\tau)&=&\\sum_{i,j}\\,\\theta^{ij}(\\tau)\\,C^{ij}(\\tau)\\\\\n",
      "&=&\\sum_{i}\\,\\theta^{ii}(\\tau)\\,C^{ii}(\\tau)+\\sum_{i \\neq j}\\,\\theta^{ij}(\\tau)\\,C^{ij}(\\tau)\\\\\n",
      "&=:&C_{split}(\\tau)+C_{herd}(\\tau).\\end{eqnarray*}"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Decomposition in pictures\n",
      "\n",
      "<h2><img src=\"http://mfe.baruch.cuny.edu/wp-content/uploads/2015/03/TothLilloFigure3.png\" align = \"center\" width=600></h2>"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Empirical observation\n",
      "\n",
      "-   Order splitting dominates herding\n",
      "\n",
      "-   There is <span>*anti-herding*</span>!\n",
      "\n",
      "    -   Other market participants tend to trade against a split meta\n",
      "        order.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Distribution of volume is fat-tailed\n",
      "\n",
      "<h2><img src=\"http://mfe.baruch.cuny.edu/wp-content/uploads/2015/03/Figure3.png\" align = \"center\" width=600></h2>"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### The Hill estimator\n",
      "\n",
      "-   Denote by $\\tilde X_{i}$, the $k$ exceedances in the sample\n",
      "    $\\{X_1,X_2,...,X_n\\}$ over some threshold $u$.\n",
      "    \n",
      "\n",
      "-   The Hill estimator (of $1/\\alpha$) is then\n",
      "    $$H_{k}=\\frac{1}{k}\\,\\sum_{i=1}^k\\,\\log \\frac{\\tilde X_{i}}{u}$$\n",
      "    \n",
      "\n",
      "-   To see how this works, consider a density of the form\n",
      "    $$f(x)=\\frac{\\alpha  }{u}\\,\\left(\\frac{u}{x}\\right)^{\\alpha +1}.$$\n",
      "    which corresponds to a distribution with tail-exponent $\\alpha$.\n",
      "    Then $$\\E\\left[\\log \\frac Xu \\right] = \\int_u^\\infty\\,\\log \\frac{x}{u}\\,f(x)\\,dx=\\frac{1}{\\alpha}.$$\n",
      "\n",
      "\n",
      "-  Hill estimator is also the maximum likelihood estimator (MLE) for the power law with pdf $f$.\n",
      "\n",
      "\n",
      "-   The Hill estimator works reasonably well when the tail is really\n",
      "    Pareto. It can give bad results if the underlying distribution does\n",
      "    not have a Pareto tail.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Distribution of volume\n",
      "\n",
      "-   The distribution of block trades appears to have a tail with\n",
      "    exponent $\\alpha \\approx 3/2$.\n",
      "\n",
      "-   Since block trading is in competition with order-splitting as a\n",
      "    strategy for executing large trades, we assume that the distribution\n",
      "    of meta orders should also have a roughly $3/2$ tail.\n",
      "\n",
      "    -   This has been confirmed recently by [Bershova and Rakhlin]<sup id=\"cite_ref-BershovaRakhlin\" class=\"reference\"><a href=\"#cite_note-BershovaRakhlin\"><span>[</span>1<span>]</span></a></sup> using a\n",
      "        database of Alliance Bernstein proprietary meta orders.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Notions of market impact\n",
      "\n",
      "The term <span>*market impact*</span> can refer to many different\n",
      "phenomena.\n",
      "\n",
      "Market impact can refer to:\n",
      "\n",
      "-   The impact of individual transactions\n",
      "\n",
      "-   The impact of meta orders\n",
      "\n",
      "-   The impact of aggregate order flow over a given period of time.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Market impact of market orders: Empirical results\n",
      "\n",
      "-   There are many studies in the classic microstructure literature that\n",
      "    document that market impact is a concave function of volume. Keim\n",
      "    and Madhavan find that block trades also generate concave market\n",
      "    impact.\n",
      "\n",
      "\n",
      "-   On the NYSE, Lillo et al. find that\n",
      "    $$\\mathbb{E} [\\Delta P|V] \\sim V^\\psi$$ with $\\psi \\approx 0.5$ for\n",
      "    small $V$ and $\\psi \\approx 0.2$ for large $V$.\n",
      "\n",
      "\n",
      "-   On the LSE, Lillo et al. find $\\psi \\approx 0.3$ on average.\n",
      "\n",
      "\n",
      "-   Potters and Bouchaud study trades on NASDAQ and the Paris Stock\n",
      "    Exchange (PSE) finding $\n",
      "    \\mathbb{E} [\\Delta P|V] \\sim \\log V.\n",
      "    $\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### $\\psi \\approx 0.3$ on LSE from Lillo et al.\n",
      "\n",
      "<h2><img src=\"http://mfe.baruch.cuny.edu/wp-content/uploads/2015/03/Figure4.png\" align = \"center\" width=600></h2>\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### The square-root formula\n",
      "\n",
      "-   Practitioners typically believe in a square-root relationship\n",
      "    between market impact and volume.\n",
      "\n",
      "\n",
      "-   Specifically, the relationship between the price impact $\\Delta P$\n",
      "    and the size $Q$ of the order should be something like\n",
      "    $$\\Delta P \\approx \\sigma\\,\\sqrt{\\frac{Q}{V}}$$ where $\\sigma$ is\n",
      "    (for example) daily volatility and $V$ is daily volume.\n",
      "\n",
      "    -   Note that this formula is dimensionally correct.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### A heuristic derivation of the square-root market impact formula\n",
      "\n",
      "-   <span>Suppose each trade impacts the mid-log-price of the stock by\n",
      "    an amount proportional to $\\sqrt{n_i}$ where $n_i$ is the size of\n",
      "    the $i$th trade.</span>\n",
      "\n",
      "\n",
      "-   <span>Then the change in mid-price over one day is given by\n",
      "    $$\\Delta P = \\sum_i^N\\,\\eta\\,\\epsilon_i\\,\\sqrt{n_i}$$ where $\\eta$\n",
      "    is the coefficient of market impact, $\\epsilon_i$ is the sign of the\n",
      "    $i$th trade and $N$ is the (random) number of trades in a day.\n",
      "    </span>\n",
      "\n",
      "    -   <span>Note that both the number of trades and the size of each\n",
      "        trade in a given time interval are random.</span>\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Derivation continued\n",
      "\n",
      "-   If $N$, $\\epsilon_i$ and $n_i$ are all independent, the variance of\n",
      "    the one-day price change is given by\n",
      "\n",
      "    \\begin{eqnarray*}\n",
      "    \\sigma^2:=\\mbox{Var}(\\Delta P)=\\eta^2\\,\\mathbb{E} [N]\\,\\mathbb{E} [n_i]=\\eta^2\\,V\\end{eqnarray*}\n",
      "\n",
      "    where $V$ is the average daily volume.\n",
      "\n",
      "\n",
      "-   <span>It follows that\n",
      "    $$|\\Delta P_i|=\\eta\\,\\sqrt{n_i}=\\sigma\\,\\sqrt{\\frac{n_i}{V}}$$ which\n",
      "    is the familiar square-root market impact formula.</span>\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Why $\\sqrt{n}$?\n",
      "\n",
      "An inventory risk argument:\n",
      "\n",
      "-   <span>A market maker requires an excess return proportional to the\n",
      "    risk of holding inventory.</span>\n",
      "\n",
      "-   <span>Risk is proportional to $\\sigma\\,\\sqrt{T}$ where $T$ is the\n",
      "    holding period.</span>\n",
      "\n",
      "-   <span>The holding period should be proportional to the size of the\n",
      "    position.</span>\n",
      "\n",
      "-   <span>So the required excess return must be proportional to\n",
      "    $\\sqrt{n}$.</span>\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Gabaix again\n",
      "\n",
      "-   Volume follows the $3/2$ law:\n",
      "    $$\\mathbb{P}\\left[\\mbox{Volume} > V\\right] \\sim \\frac{1}{V^{3/2}}$$\n",
      "\n",
      "\n",
      "-   Market impact is proportional to the square-root of volume\n",
      "    $$\\Delta P \\sim \\sqrt{V}$$\n",
      "\n",
      "\n",
      "-   Then $$\\mathbb{P}\\left[\\Delta P > x\\right] \\sim \\frac{1}{x^3},$$ the so-called cubic\n",
      "    law of returns.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Aggregate market impact\n",
      "\n",
      "-   When practitioners are questioned as to whether the square-root\n",
      "    model is supposed to estimate the impact of individual orders, meta\n",
      "    orders or aggregate order flow, the answer is usually \u201call of\n",
      "    these!\u201d.\n",
      "\n",
      "-   [Hopman]<sup id=\"cite_ref-Hopman\" class=\"reference\"><a href=\"#cite_note-Hopman\"><span>[</span>5<span>]</span></a></sup> measures aggregate order flow on the PSE as\n",
      "    $\\sum_i\\,\\epsilon_i\\,V_i^\\psi$.\n",
      "\n",
      "    -   He finds that $\\psi = 1/2$ best explains daily returns. $\\psi=1$\n",
      "        gives a much lower $R^2$.\n",
      "\n",
      "    -   He also finds that causality runs from orders to prices and not\n",
      "        the other way round, supporting the econophysicists and\n",
      "        contradicting the informed trader story.\n",
      "\n",
      "-   Gabaix et al. find $\\psi \\approx 1/2$ on New York, London and Paris\n",
      "    markets.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Scaling of aggregated returns\n",
      "\n",
      "<h2><img src=\"http://mfe.baruch.cuny.edu/wp-content/uploads/2015/03/Figure5.png\" align = \"center\" width=600></h2>\n",
      "<a name=\"fig5\"></a>"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Aggregation\n",
      "\n",
      "-   For a sequence of $N$ successive trades, let\n",
      "    $Q_N=\\sum_i^N \\epsilon_i\\,V_i$ be the aggregate quantity and\n",
      "    $R_N=\\sum_i^N\\,\\log P_i/P_{i-1}$ be the aggregate return.\n",
      "\n",
      "-   [Figure 5](#fig5) shows how the aggregate return $R(Q,N)$ scales with\n",
      "    quantity $Q_N$ and $N$.\n",
      "\n",
      "-   We see that according to [Bouchaud, Farmer, Lillo]<sup id=\"cite_ref-BouchaudFarmerLillo\" class=\"reference\"><a href=\"#cite_note-BouchaudFarmerLillo\"><span>[</span>2<span>]</span></a></sup>, aggregate impact\n",
      "    becomes increasingly linear with increasing $N$.\n",
      "\n",
      "    -   Note however that this seems to contradict Hopman.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Why is market impact concave?\n",
      "\n",
      "There are three types of answer in the literature\n",
      "\n",
      "1.  The informativeness of a trade depends on size (Easley and O\u2019Hara).\n",
      "\n",
      "    -   Small trades may carry almost as much information as large\n",
      "        trades.\n",
      "\n",
      "2.  The shape of the order book (Weber and Rosenow).\n",
      "\n",
      "    -   The cumulative depth available at a given price level determines\n",
      "        price impact.\n",
      "\n",
      "3.  Selective liquidity taking.\n",
      "\n",
      "    -   Traders condition their orders on the quantity available in the\n",
      "        order book.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Virtual market impact\n",
      "<h2><img src=\"http://mfe.baruch.cuny.edu/wp-content/uploads/2015/03/Figure6.png\" align = \"center\" width=600></h2>"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Selective liquidity taking\n",
      "\n",
      "-   Empirically, nearly all trades are of sizes less than or equal to\n",
      "    the size available at the best quote, and certainly smaller than the\n",
      "    total size available at the first two quote levels.\n",
      "\n",
      "\n",
      "-   Suppose then that the price changes only in increments of the spread\n",
      "    $s$. Then\n",
      "    $$\\mathbb{E} [\\Delta P|V]=\\Pr(\\Delta P >0|V)\\,s=\\Pr(V \\ge Q)\\,s$$\n",
      "    where $Q$ is the quantity available at the best quote.\n",
      "\n",
      "\n",
      "-   But $$\\Pr(V \\ge Q)=\\Pr(Q \\le V)=:F_Q(V),$$ the cumulative\n",
      "    distribution function of quantity at the best quote, which we expect\n",
      "    to be concave wrt V.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Gamma-distributed volume at best quote\n",
      "\n",
      "For example, according to Bouchaud, M\u00e9zard and Potters, the volume at\n",
      "the best quote is gamma-distributed\n",
      "$$f(V) \\sim V^{\\gamma-1}\\,e^{-V/V_0}$$ with $\\gamma \\approx 0.75$.\n",
      "\n",
      "The cumulative distribution would then be\n",
      "$$F(V)=1-\\frac{\\Gamma \\left(\\gamma ,\\frac{V}{V_0} \\right)}{\\Gamma (\\gamma)}$$\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Gamma distributed volume at best quote with $\\gamma=0.75$\n",
      "\n",
      "<h2><img src=\"http://mfe.baruch.cuny.edu/wp-content/uploads/2015/03/FV.png\" align = \"center\" width=600></h2>\n",
      "\n",
      "The green line is $F(V)$; the red line is $V^{0.5}$; the violet line is\n",
      "$V^{0.2}$.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### A fixed permanent impact model\n",
      "\n",
      "-   Suppose that changes in the mid-quote $m$ depend on the sign\n",
      "    $\\epsilon$ of the order, the order size $v$ and possibly on other\n",
      "    state variables $\\Omega$ such as the state of the order book:\n",
      "    $$\\Delta m_t=m_{t}-m_{t-1}=\\epsilon_t\\,f(v_t,\\Omega_t)+\\eta_t$$ with\n",
      "    $\\eta$ iid and satisfying $\\mathbb{E} [\\eta_t]=0$,\n",
      "    $\\mbox{Var}[\\eta_t]=\\sigma^2$.\n",
      "\n",
      "\n",
      "-   Integrating this gives\n",
      "    $$m_{t}=\\sum_{k<t}\\,\\epsilon_k\\,f(v_k,\\Omega_k)+\\sum_{k<t}\\,\\eta_k$$\n",
      "    from which we see that in this model, all market impact is\n",
      "    permanent.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Variance and autocovariance of price returns\n",
      "\n",
      "-   If $f(\\cdot)$ is independent of order-flow, it follows that\n",
      "\n",
      "    $$\\mathbb{E} [\\Delta m_{t}\\,\\Delta m_{t+\\tau}]=\\mathbb{E} [\\epsilon_{t}\\,\\epsilon_{t+\\tau}\\,f(v_t)\\,f(v_{t+\\tau})]\\propto \\tau^{-\\alpha}$$\n",
      "    \n",
      "    so that price returns are autocorrelated in time.\n",
      "\n",
      "\n",
      "-   Also, in that case, the variance of lag-$\\tau$ returns is given by\n",
      "    $$\\mathbb{E} [(m_{t+\\tau}-m_t)^2]\\propto \\sum_{i,j=1}^\\tau\\,\\mathbb{E} \\left[\\epsilon_i\\,\\epsilon_j\\right]\\,f(v_i)\\,f(v_j) \\propto \\tau^{2-\\alpha}.$$\n",
      "\n",
      "\n",
      "-   Both of these properties of the model are inconsistent with\n",
      "    efficient prices as empirically observed.\n",
      "\n",
      "\n",
      "-   It follows that either the market impact function $f(\\cdot)$ must\n",
      "    depend on order flow or that market impact is not permanent.\n",
      "\n",
      "    -   The former point of view was promoted by Farmer, Lillo et al.\n",
      "\n",
      "    -   The latter point of view was promoted by Bouchaud et al.\n",
      "\n",
      "\n",
      "-   We will see that, somewhat surprisingly, the two points of view are compatible with each other.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### The Madhavan Richardson and Roomans (MRR) model again\n",
      "\n",
      "-   In the MRR model, as in Glosten-Milgrom, the revision in beliefs is\n",
      "    positively correlated with the innovation in the order flow:\n",
      "\n",
      "    <a name=\"eq:MRRsurprise\"></a>(4)\n",
      "    $$\\Delta V_t=\\lambda\\,\\left(\\epsilon_t-\\mathbb{E} [\\epsilon_t|\\mathcal{F} _t]\\right) + e_t$$\n",
      "\n",
      "    where $V_t$ is the efficient price and $e_t$ represents for example\n",
      "    news.\n",
      "\n",
      "\n",
      "-   However, in the MRR model, the revision in beliefs depends only on\n",
      "    the unexpected component of order flow.\n",
      "\n",
      "\n",
      "-   MRR modeled the order flow as $AR(1)$: $\\epsilon_t = \\phi_1 \\epsilon_{t-1} + \\eta_t$.\n",
      "\n",
      "\n",
      "-   We now model order flow as a long-memory process.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Expected trade sign\n",
      "\n",
      "-   In the MRR model,\n",
      "    $$\\mathbb{E} [\\epsilon_t|\\epsilon_{t-1}]=\\phi_1\\,\\epsilon_{t-1}.$$\n",
      "    where $\\phi_1$ is the first order autocorrelation coefficient.\n",
      "\n",
      "\n",
      "-   In our extension of this model, we write\n",
      "    $$\\hat \\epsilon_t=\\mathbb{E} [\\epsilon_t|\\mathcal{F} _t]=\\sum_{k=1}^\\infty\\,\\phi_k\\,\\epsilon_{t-k}$$\n",
      "    so the filtration $\\mathcal{F} _t$ is now not just the last order\n",
      "    sign $\\epsilon_{t-1}$ but the entire order flow history.\n",
      "\n",
      "\n",
      "-   In practice, we can fit an $AR(p)$ model with some large order $p$:\n",
      "    $$\\hat \\epsilon_t=\\mathbb{E} [\\epsilon_t|\\mathcal{F} _t]=\\sum_{k=1}^p\\,\\phi_k\\,\\epsilon_{t-k}.$$\n",
      "\n",
      "\n",
      "-   We expect the coefficients to decay as a power-law:\n",
      "    $$\\phi_{k}\\sim k^{-\\beta} \\text{ for some $\\beta$.}$$\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Market impact relates to unexpected order flow\n",
      "<h2><img src=\"http://mfe.baruch.cuny.edu/wp-content/uploads/2015/03/Figure10.png\" align = \"center\" width=600></h2>\n",
      "<a name=\"fig10\"></a>"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Empirical verification of the model\n",
      "\n",
      "-   Approximating $\\Delta V$ by $\\Delta m$, we may rewrite [(4)](#eq:MRRsurprise) as\n",
      "\n",
      "    <a name=\"eq:MRRe\"></a>(5)\n",
      "    $$\\Delta m_{t}=\\lambda\\,\\left(\\epsilon_t-\\hat \\epsilon_t\\right)+\\eta_t$$\n",
      "\n",
      "\n",
      "-   In this model, the impact of a transaction depends on the entire\n",
      "    history of the order flow and how predictable the order sign of the\n",
      "    transaction is.\n",
      "\n",
      "    -   The most likely outcome has the smallest impact.\n",
      "\n",
      "\n",
      "-   In [Figure 10](#fig10), we see stunning empirical verification of this model.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Bouchaud\u2019s power-law decay argument\n",
      "\n",
      "-   <span> As before, assume that over one day\n",
      "    $$\\Delta P = \\sum_i^N\\,\\eta\\,\\epsilon_i\\,\\sqrt{n_i}$$</span>\n",
      "\n",
      "-   <span>The previous heuristic proof of the square-root model assumed\n",
      "    that $\\mbox{Cov}[\\epsilon_i,\\epsilon_j]=0$ if $i \\neq j$ and that all\n",
      "    market impact is permanent. </span>\n",
      "\n",
      "-   <span> Empirically, we found that autocorrelation of trade signs\n",
      "    shows power-law decay with a small exponent $\\alpha$ (very slow\n",
      "    decay).</span>\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Bouchaud\u2019s power-law decay argument continued\n",
      "\n",
      "\\begin{eqnarray*}\n",
      "\\mbox{Var}[\\Delta P] &=& \\eta^2\\,\\mbox{Var}\\left[\\sum_i^N\\,\\epsilon_i\\,\\sqrt{n_i}\\right]\\\\\n",
      "&=&\\eta^2\\,\\left\\{N\\,\\mbox{Var}[\\sqrt{n_i}]+\\sum_{i \\neq j}\\,\\mbox{Cov}[\\epsilon_i,\\epsilon_j]\\right\\}\\\\\n",
      "&\\approx& \\eta^2\\,\\left\\{N\\,\\mbox{Var}[\\sqrt{n_i}]+\\frac{2\\,C_1}{(2-\\alpha)\\,(1-\\alpha)}\\,\\mathbb{E} [\\sqrt{n}]^2\\,N^{2-\\alpha}\\right\\}\\\\\n",
      "&\\sim& N^{2-\\alpha} \\textrm{ as } N \\to \\infty\\end{eqnarray*}\n",
      "\n",
      "-   <span>Empirically, we find that, to a very good approximation,\n",
      "    $\\mbox{Var}[\\Delta P] \\propto N$.</span>\n",
      "\n",
      "    -   <span>Otherwise returns would be serially correlated.</span>\n",
      "\n",
      "-   <span>These observations may be reconciled if market impact decays\n",
      "    as a power law.</span>\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Computation of daily variance with power-law decay\n",
      "\n",
      "-   Assuming market impact decays as $1/\\tau^{\\gamma}$, we have\n",
      "\n",
      "    \\begin{eqnarray*}\n",
      "    \\mbox{Var}[\\Delta P] &=& \\eta^2\\,\\mbox{Var}\\left[\\sum_i^N\\,\\frac{\\epsilon_i\\,\\sqrt{n_i}}{(N-i)^{\\gamma}}\\right]\\\\\n",
      "    &=&\\eta^2\\,\\left\\{\\sum_i^{N-1}\\,\\frac{\\mathbb{E} [n]}{(N-i)^{2\\,\\gamma}} + 2\\,C_1\\, \\sum_{i=1}^{N-1}\\,\\sum_{j=i+1}^{N-1}\\frac{\\mathbb{E} [\\sqrt{n}]^2}{(N-i)^{\\gamma}\\,(N-j)^{\\gamma}\\,(j-i)^{\\alpha}}\\right\\}\\\\\n",
      "    &\\sim& N^{2-\\alpha-2\\,\\gamma} \\textrm{ as } N \\to \\infty\\\\\n",
      "    & \\sim & N \\textrm{ only if } \\gamma \\approx (1-\\alpha)/2.\\end{eqnarray*}\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Equivalence of the two formulations\n",
      "\n",
      "We have in the Lillo picture that\n",
      "\n",
      "<a name=\"eq:dm1\"></a>(6)\n",
      "$$\\Delta m_t=m_t-m_{t-1}=\\lambda\\,\\left(\\epsilon_t-\\hat \\epsilon_t\\right)+\\eta_t$$\n",
      "\n",
      "with $$\\hat \\epsilon_t=\\sum_{k=1}^\\infty\\,\\phi_k\\,\\epsilon_{t-k}.$$\n",
      "Write (Bouchaud picture)\n",
      "$$m_t=\\lambda\\,\\sum_{i=0}^\\infty\\,G(i)\\,\\epsilon_{t-i}+\\sum_{j \\leq t}\\,\\eta_j$$\n",
      "Then\n",
      "\n",
      "<a name=\"eq:dm2\"></a>(7)\n",
      "\\begin{eqnarray*}\n",
      "m_t-m_{t-1}&=&\\lambda\\,\\left(\\sum_{k=0}^\\infty\\,G(k)\\,\\epsilon_{t-k}-\\sum_{k=0}^\\infty\\,G(k)\\,\\epsilon_{t-1-k}\\right)+\\eta_t\\nonumber\\\\\n",
      "&=&\\lambda\\,\\epsilon_t+\\lambda\\,\\sum_{k=1}^\\infty\\,\\left\\{G(k)-G(k-1)\\right)\\}\\,\\epsilon_{t-k}+\\eta_t\n",
      "\\end{eqnarray*}"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "To match the expressions [(6)](#eq:dm1) and [(7)](#eq:dm2), we need $$G(k)-G(k-1)=-\\phi_k.$$ One\n",
      "choice that would work is: $$G(k)=\\sum_{j=k+1}^{\\infty}\\,\\phi_j$$ If the\n",
      "autocorrelation function $\\rho(\\tau) \\sim \\tau^{-\\alpha}$ for large\n",
      "$\\tau$, and if the underlying process for order signs is ARFIMA, the\n",
      "best linear predictor of order sign is given by\n",
      "$$\\hat \\epsilon_t=\\sum_{k=1}^\\infty\\,\\phi_k\\,\\epsilon_{t-k}$$ with\n",
      "$\\phi_k \\sim k^{-(1+\\beta)}$ for large $k$ and\n",
      "$$\\beta=\\frac{1-\\alpha}{2}.$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "Then, for large $k$,\n",
      "$$G(k)=\\sum_{j=k+1}^{\\infty}\\,\\phi_j \\sim \\frac{1}{k^\\beta}$$ with\n",
      "$$\\beta=\\frac{1-\\alpha}{2}.$$ That is, the propagator $G(\\tau)$ decays\n",
      "as $\\tau^{-\\gamma}$ with $$\\gamma=\\frac{1-\\alpha}{2}.$$ as required to\n",
      "show equivalence between the Bouchaud and Lillo pictures."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "### Explicit computation \n",
      "\n",
      "Suppose the order sign process is $ARFIMA(0,d,0)$. Then, by definition,\n",
      "$$(1-B)^d \\epsilon_t=\\eta_t$$ where $\\eta_t \\sim N(0,1)$ say.\n",
      "\n",
      "Inverting this expression gives\n",
      "\n",
      "<a name=\"eq:epsilonEta\"></a>(8)\n",
      "$$\\epsilon_t=\\eta_t+\\sum_{j=1}^\\infty\\,\\psi_j\\,\\eta_{t-j}=\\sum_{j=0}^\\infty\\,\\psi_j\\,\\eta_{t-j}$$\n",
      "\n",
      "where we adopt the convention that $\\psi_0=1$ and for $j>0$,\n",
      "\n",
      "$$\\psi_j=(-1)^j\\,\\left(\n",
      "\\begin{array}{c}-d\\\\j\\end{array}\\right)\n",
      "=\\frac{d\\,(d+1)...\\,(d+j-1)}{j!}\n",
      "=\\left(\n",
      "\\begin{array}{c}j+d-1\\\\j\\end{array}\\right)\n",
      "$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "The autocovariance function of $\\epsilon_t$ is then\n",
      "\n",
      "<a name=\"eq:arfimaACF\"></a>(9)\n",
      "\\begin{eqnarray*}\n",
      "\\rho(k)&=&\\sum_{j=0}^\\infty\\,\\psi_j\\,\\psi_{j+k}\\nonumber\\\\\n",
      "&=&\\frac{\\Gamma(1-2\\,d)}{\\{\\Gamma(1-d)\\}^2}\\,\\frac{d\\,(1+d)...\\,(k-1+d)}{(1-d)\\,(2-d)\\,...\\,(k-d)}\\nonumber\\\\\n",
      "&\\sim&\\frac{\\Gamma(1-2\\,d)}{\\Gamma(d)\\,\\Gamma(1-d)}\\,\\frac{1}{k^{1-2\\,d}} ~ \\text{ as } k \\to \\infty\n",
      "\\end{eqnarray*}\n",
      "\n",
      "In our notation, $\\rho(\\tau) \\sim \\tau^{-\\alpha}$ so\n",
      "$d=\\frac{1-\\alpha}{2}$.\n",
      "\n",
      "Now invert [(8)](#eq:epsilonEta) to get\n",
      "$$\\epsilon_t=\\eta_t+\\sum_{j=1}^\\infty\\,(-1)^{j-1}\\,\\left(\n",
      "\\begin{array}{c}d\\\\j\n",
      "\\end{array}\n",
      "\\right)\\,\\epsilon_{t-j}.$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "Then\n",
      "$$\\hat \\epsilon_t=\\mathbb{E} [\\epsilon_t|\\mathcal{F} _t]=\\sum_{k=1}^p\\,\\phi_k\\,\\epsilon_{t-k}. \\text{ with }\n",
      "\\phi_k=(-1)^{j-1}\\,\\left(\n",
      "\\begin{array}{c}d\\\\j\n",
      "\\end{array}\n",
      "\\right).$$ Finally $$G(k)=\\sum_{j=k+1}^\\infty\\,\\phi_j=\\left(\n",
      "\\begin{array}{c}k-d\\\\k\n",
      "\\end{array}\n",
      "\\right)\\sim \\frac{1}{\\Gamma(1-d)\\,k^d} \\text{ as } k \\to \\infty.$$ Thus,\n",
      "the exponent $\\gamma$ of decay of market impact is given by\n",
      "$$\\gamma=d=\\frac{1-\\alpha}{2}.$$\n",
      "\n",
      "<font color=\"red\">The Bouchaud and Lillo formulations are exactly equivalent in this case."
     ]
    },
    {
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      "### Including other events\n",
      "\n",
      "Start with the Bouchaud transient impact (TIM) picture again:\n",
      "$$p_t=\\lambda\\,\\sum_{s<t}\\,[G(t-s)\\,\\epsilon_s +\\eta_s]+p_{-\\infty}.$$\n",
      "The <span>*propagator*</span> or <span>*decay kernel*</span> $G(\\cdot)$\n",
      "may be estimated from the correlation function\n",
      "\n",
      "\\begin{eqnarray*}\n",
      "\\mathcal{R} (\\ell)&=&\\langle (p_{t+\\ell}-p_t)\\,\\epsilon_t  \\rangle\\\\\n",
      "&=&\\sum_{0 < n \\leq \\ell}\\,G(n)\\,C(\\ell - n) + \\sum_{n>0}\\left[G(n + \\ell) - G(n)\\right]\\,C(n).\\end{eqnarray*}\n",
      "\n",
      "Then, the second moment of the price difference (variance) may be\n",
      "computed as $$\\mathcal{D} (\\ell)=\\langle (p_{t+\\ell}-p_t)^2\\rangle.$$\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
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       "slide_type": "slide"
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     "source": [
      "### Bouchaud transient impact picture vs observation\n",
      "\n",
      "[Eisler, Bouchaud and Kockelkoren]<sup id=\"cite_ref-EislerEtAl\" class=\"reference\"><a href=\"#cite_note-EislerEtAl\"><span>[</span>4<span>]</span></a></sup> show that the Bouchaud transient impact picture is\n",
      "incomplete.\n",
      "\n",
      "<h2><img src=\"http://mfe.baruch.cuny.edu/wp-content/uploads/2015/03/EislerEtAlFigure1.png\" align = \"center\" width=600></h2>"
     ]
    },
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      "### Interpretation\n",
      "\n",
      "The Bouchaud picture is too simplistic\n",
      "\n",
      "-   Market impact varies wildly over time according to the state of the\n",
      "    market (for example the shape of the order book)\n",
      "\n",
      "    -   The history of order flow is not the only determinant of market\n",
      "        impact\n",
      "\n",
      "-   Not only market orders impact the market price; limit orders and\n",
      "    cancelations also impact the price.\n",
      "\n"
     ]
    },
    {
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     "metadata": {
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      "### Summary\n",
      "\n",
      "-   Order flow is a long memory process.\n",
      "\n",
      "    -   The dominant effect is order-splitting.\n",
      "\n",
      "-   Market impact is concave due to selective liquidity taking.\n",
      "\n",
      "-   Market impact of market orders can be modeled as:\n",
      "\n",
      "    -   Permanent but state-dependent (Lillo)\n",
      "\n",
      "    -   Transient (Bouchaud)\n",
      "\n",
      "-   Both of these formulations are equivalent.\n",
      "\n",
      "-   To get quantitative (as opposed to qualitative) agreement with\n",
      "    observation, we need to take into account\n",
      "\n",
      "    -   Time-varying liquidity\n",
      "\n",
      "    -   Limit orders and cancelations\n",
      "\n"
     ]
    },
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     "source": [
      "### References\n",
      "<br />\n",
      "\n",
      "<div class=\"reflist\" style=\"list-style-type: decimal;\">\n",
      "\n",
      "<ol>\n",
      "\n",
      "  <li id=\"cite_note-BershovaRakhlin\"><span class=\"mw-cite-backlink\"><b><a href=\"#cite_ref-BershovaRakhlin\">^</a></b></span>Nataliya Bershova and Dmitry Rakhlin, The\n",
      "Non-Linear Market Impact of Large Trades: Evidence from Buy-Side Order\n",
      "Flow, <span>*Quantitative Finance*</span> <span>**13**</span>(11)\n",
      "1759\u20131778 (2013). \n",
      "  </li>\n",
      "\n",
      "  <li id=\"cite_note-BouchaudFarmerLillo\"><span class=\"mw-cite-backlink\"><b><a href=\"#cite_ref-BouchaudFarmerLillo\">^</a></b></span>Jean-Philippe Bouchaud, J. Doyne Farmer, and\n",
      "Fabrizio Lillo, How Markets Slowly Digest Changes in Supply and Demand,\n",
      "in <span>*Handbook of Financial Markets: Dynamics and Evolution*</span>\n",
      "57\u2013156. (2009) available at\n",
      "http://tuvalu.santafe.edu/%7Ejdf/papers/MarketsSlowlyDigest.pdf:\n",
      "Sections 4, 5 and 6. \n",
      "  </li>\n",
      "    \n",
      "  <li id=\"cite_note-BouchaudEtAl\"><span class=\"mw-cite-backlink\"><b><a href=\"#cite_ref-BouchaudEtAl\">^</a></b></span>Jean-Philippe Bouchaud, Yuval Gefen, Marc Potters and Matthieu Wyart, Fluctuations and response in financial markets: the subtle nature of random price changes, <span>*Quantitative Finance*<span> <span>**4**</span>(2) 176--190 (2004). \n",
      "  </li>\n",
      "  \n",
      "  <li id=\"cite_note-EislerEtAl\"><span class=\"mw-cite-backlink\"><b><a href=\"#cite_ref-EislerEtAl\">^</a></b></span>Zoltan Eisler, Jean-Philippe\n",
      "Bouchaud, and Julien Kockelkoren, Models for the impact of all order\n",
      "book events, <span>*arXiv.org*</span> (July 18, 2011).\n",
      "  </li>\n",
      "  \n",
      "  <li id=\"cite_note-Hopman\"><span class=\"mw-cite-backlink\"><b><a href=\"#cite_ref-Hopman\">^</a></b></span>Carl Hopman, Do supply and demand drive stock prices?,\n",
      "<span>*Quantitative Finance*</span> <span>**7**</span>(1) 37\u201353 (2007). \n",
      "  </li>\n",
      "  \n",
      "  <li id=\"cite_note-LilloMikeFarmer\"><span class=\"mw-cite-backlink\"><b><a href=\"#cite_ref-LilloMikeFarmer\">^</a></b></span>Fabrizio Lillo, Szabolcs Mike, and J Doyne\n",
      "Farmer, Theory for long memory in supply and demand, <span>*Phys. Rev.\n",
      "E*</span> <span>**71**</span>(6) 66122 (2005). \n",
      "  </li>\n",
      "  \n",
      "  <li id=\"cite_note-SadTaqqu\"><span class=\"mw-cite-backlink\"><b><a href=\"#cite_ref-SadTaqqu\">^</a></b></span>Gennady Samorodnitsky, Murad S. Taqqu,\n",
      "<span>*Stable non-Gaussian random processes: stochastic models with\n",
      "infinite variance*</span>, Chapman and Hall (1994). \n",
      "  </li>\n",
      "  \n",
      "  <li id=\"cite_note-TothLillo\"><span class=\"mw-cite-backlink\"><b><a href=\"#cite_ref-TothLillo\">^</a></b></span>Bence T\u00f3th, Imon Palit, Fabrizio Lillo, and J Doyne\n",
      "Farmer, Why is order flow so persistent?, <span>*arXiv*</span> available\n",
      "at <http://arxiv.org/abs/1108.1632v1> (2011). \n",
      "  </li>\n",
      "  \n",
      "</ol>"
     ]
    }
   ],
   "metadata": {}
  }
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}